Autioning Supply Contrats By Fangruo Chen June 6, 2001 Revised: May 7, 2004 DRO-2004-02 Deision, Risk & Operations Working Papers Series
Autioning Supply Contrats Fangruo Chen Graduate Shool of Business Columbia University New York, NY 10027 June 6, 2001 This version: May 7, 2004
Abstrat This paper studies a prourement problem where both the purhase quantity and the prie need to be determined. There are multiple potential suppliers with private information about their own prodution osts. An optimal prourement strategy involves the autioning o of a supply ontrat that spei es a transfer payment for eah possible purhase quantity and the delegation of the quantity deision to the seleted supplier. In the ontext of a newsvendor model with supply-side ompetition, the paper establishes a onnetion between the optimal prourement strategy and a ommon pratie in the retail industry, namely the use of slotting allowanes and vendor managed inventory. Also disussed in the newsvendor ontext are the role of well-known supply ontrats suh as returns ontrats and revenuesharing ontrats in prourement autions, the senarios where the suppliers may have more or less information about the demand distribution vis-a-vis the prourer, and the ost of demand unertainty and how it hanges as the supply-side ompetition inreases. The optimal prourement strategy is also ompared to a suboptimal, but simpler, strategy where the prourer rst determines a purhase quantity and then seeks the lowest ost supplier for the quantity in an aution.
1 Introdution Mathematial models in operations management addressing prourement deisions (i.e., when and how muh to buy) often make the simplisti assumption that the pries for the items to be purhased are exogenously given. For example, the elebrated newsvendor model typially assumes a given purhase prie. The plethora of inventory models in the literature also make a similar assumption. In reality, prourement managers often need to disover the pries for the items they want to buy, and the disovery proess often involves market researh, negotiations, autions/bidding, et. Prourement models in eonomis, on the other hand, tend to fous on prie disovery, xing quantity deisions. The purpose of this paper is to understand how prie disovery an be integrated with quantity deisions toreahatanoptimalprourementstrategy. We onsider a model where a rm's prourement manager faes a number of potential suppliers. The question is how muh of a produt should be purhased and from whih supplier. The rm'srevenueisaonave,inreasingfuntionofthepurhasequantity. Eah supplier an produe the produt at a onstant marginal ost, and without any apaity limit. The marginal osts of di erent suppliers all ome from a ommon probability distribution, with eah supplier privately informed about his own marginal ost. The suppliers are risk neutral. The prourement manager's objetive is to maximize the rm's expeted pro t, whih is the rm's revenue (as a funtion of the purhase quantity) minus the prourement ost. The strategy that leads to the maximum pro t is an optimal prourement strategy, and its struture shows how prie disovery should be integrated with the quantity deision. A key result of the paper is that an optimal prourement strategy takes the form of an entry-fee aution. Here the rm rst ommits to a supply ontrat, whih spei es a payment for eah possible quantity that the seleted supplier may deliver. Eah potential supplier views this supply ontrat as a business opportunity, taking the payment funtion o ered by the rm as their revenue funtion. Based on their own marginal prodution ost, they eah determine an optimal quantity to produe and deliver to the rm to maximize their own pro t. Note that di erent suppliers derive di erent values from the supply ontrat, with the lowest-marginal-ost supplier deriving the most value. The suppliers ompete for the supply ontrat in an aution by submitting an entry fee they are willing to pay, with the winner being the supplier o ering the highest entry fee. 1 Theautionanbeonduted 1 An alternative way to implement an entry-fee aution is to require eah supplier to not only submit an 1
in many di erent forms, inluding the English aution, the Duth aution, the rst-prie, sealed-bid aution, or the Vikrey aution. With a properly hosen supply ontrat, the entry-fee aution ahieves the maximum expeted pro t for the rm among all possible prourement strategies. The integration of prie disovery and the quantity deision therefore requires the design of a supply ontrat (a quantity-to-payment mapping) followed by an aution that awards the ontrat to a supplier, who is given the deision right to determine the purhase quantity. This is intuitive. The aution hooses the most e±ient supplier. Moreover, the lowest marginal ost among the suppliers together with the supply ontrat o ered by the rm determine the ultimate quantity deision. In other words, the quantity deision omes from both sides of the trade. Entry-fee autions t well with a prevalent pratie in the retail industry, i.e., the use of slotting allowanes and vendor managed inventory (VMI). A slotting allowane is an upfront, lump-sum fee that a manufaturer pays to a retail store when it introdues a new produt to the store. It an also be a fee paid in order to keep an existing produt in the store, although in this ase, it is often referred to as a pay-to-stay fee. The total of slotting allowanes is estimated at between $6 and $9 billion a year (Rao and Mahi 2003). The reasons given for using slotting allowanes often fous on the need to signal to the retailer the manufaturer's belief about the produt's market potential or the need to ompensate the retailer for the risks of arrying a new produt. On the other hand, VMI refers to the pratie of delegating inventory deisions to the supplier. The arguments for this re-alloation of stoking deision rights are multi-faeted, inluding the bargaining power of the retailer (e.g., Wal-Mart), the vendor's expertise in managing inventories, the vendor's ability to oordinate the prodution and distribution of multiple produts, modern ommuniation tehnologies, et. While VMI has been hailed as an important strategy for improving supply hain e±ieny, the use of slotting allowanes is more ontroversial, attrating antitrust inquiries (see, e.g., FTC Testimony (1999)). It is therefore interesting that an optimal prourement strategy identi ed in this paper, i.e., the entry-fee autions, atually requires the use of a slotting allowane (i.e., the entry fee) and the delegation of the prodution/inventory deision to the suppliers. The fee results from the suppliers' ompetition for the retailer's shelf spae, while the alloation of deision rights is to take entry-fee o er but also to speify his delivery quantity. So eah bid has two dimensions, a fee and a quantity. But the winner is hosen solely based on the entry fee. This implementation eliminates any unertainty about what a supplier may deliver after being hosen. 2
advantage of the suppliers' private ost information. Our results have the potential to ontribute to the on-going debate on the purposes and onsequenes of slotting allowanes. Interpreting the rm in our prourement model as a newsvendor faing an inventory deision at the beginning of a selling season with unertain demand, we have a newsvendor model with supply-side ompetition. Under this interpretation, we an make the supply ontrat used in an entry-fee aution demand-dependent, suh as in a returns ontrat or a revenue-sharing ontrat. A returns ontrat alls for a wholesale prie paid for eah unit delivered to the newsvendor at the beginning of the selling season, and a rebate for eah unit of exess inventory at the end of the season. On the other hand, a revenuesharing ontrat ontains a wholesale prie for eah unit of inventory at the beginning of the selling season, and an agreement on how the realized revenues are to be split between the newsvendor and the seleted supplier. Note that under either a returns ontrat or a revenue-sharing ontrat, the transfer payment for any given purhase quantity (or initial inventory) is unertain (beause it depends on the realized demand). This is learly different from the ontrat onsidered so far, whereby the transfer payment is a deterministi funtion of the purhase quantity. But for our risk-neutral suppliers, all that matters is the expeted transfer payment as a funtion of the purhase quantity. It is shown that under fairly general onditions, a returns ontrat or a revenue-sharing ontrat an indeed be foundthat,whenoupledwithanautionmehanism,onstitutesanoptimalprourement strategy for the rm. We also disuss the advantages and disadvantages of these demanddependent ontrats if the newsvendor and the suppliers have asymmetri information about the probability distribution of the demand. An alternative prourement strategy is for the rm to hoose a purhase quantity and then run an aution to identify a supplier that an deliver the quantity at the lowest ost. We refer to this strategy as a xed-quantity aution. It is suboptimal beause the quantity deision only takes into aount the distributional knowledge the rm has about the suppliers' osts (the atual prodution osts do not play a role in the quantity deision). But it is simpler than the optimal strategy, beause it does not involve the spei ation of a payment funtion. There are two deision variables that need to be determined: the purhase quantity and the reserve prie for the aution. For the newsvendor setting, we develop an algorithm to determine the optimal values of these variables (that maximize the rm's expeted pro t). Numerial examples are used to illustrate the di erene in the rm's expeted pro t between the xed-quantity aution and the optimal prourement strategy. 3
Diretly related to this paper is Dasgupta and Spulber (1989/90), who have provided a di erent solution to the above prourement problem. We refer to their solution as quantity autions, whih works as follows. As in entry-fee autions, the rm rst announes a supply ontrat (a payment funtion). The suppliers then submit quantity o ers in a sealed, high-bid aution. The supplier with the highest bid (i.e., quantity o er) wins the aution, produes and delivers his bid, and reeives a payment aording to the pre-announed ontrat. A quantity aution with a properly hosen supply ontrat maximizes the rm's expeted pro t. But unlike entry-fee autions, quantity autions must be onduted in the sealed-bid fashion, and would lose their optimality when implemented in other formats. Another distintion between quantity autions and entry-fee autions is the amount of detail required to determine the optimal payment funtion: quantity autions require the number of potential suppliers, whereas entry-fee autions do not. 2 Also losely related to this paper is the literature on multidimensional autions, see, e.g., Che (1993) and Brano (1997). Here the suppliers ompete for a prourement ontrat in terms of not only prie but also quality. For example, the Department of Defense, in prouring a weapons system, ares about the system's performane as well as prie. The optimal design of a multidimensional aution typially spei es a soring rule that ombines the multiple dimensions of a bid (e.g., a quality spei ation and a prie) into a single sore, whih is then used to determine a winner. Note that the prourement ompetition studied in this paper also has two dimensions: prie and quantity (instead of quality). Our optimal designanalsobeviewedasasoringrule. Forexample,inanentry-feeaution,the suppliers eah submit an entry fee and a quantity, and the soring rule puts all the weight on the entry-fee dimension. It is therefore interesting that this design, when supported by a properly hosen supply ontrat, an ahieve optimality for the prourer. This paper is at the intersetion of aution theory and operations management. Aution theory has grown tremendously sine Vikrey's (1961) seminal work; MAfee and MMillan (1987a) and Klemperer (1999) provide omprehensive surveys of the theory. Studies of supply ontrats have reently reeived muh researh attention in operations management. 2 An alternative way to implement a quantity aution is via a wholesale prie aution. Here the rm rst announes a purhasing plan, whih spei es the quantity the rm is ommitted to purhase as a funtion of the prevailing wholesale prie. The suppliers then engage in a sealed, low-bid aution, where the supplier submitting the lowest wholesale prie wins. The transation is ompleted at the lowest bid (wholesale prie) and the orresponding purhase quantity. The purhasing plan an be shown to be a quantity disount plan, i.e., the purhase quantity inreases as the wholesale prie dereases. Wholesale prie autions have been studied by Hansen (1988) and Jin and Wu (2000), where the purhasing plan is exogenously given. 4
The fous is the impat of di erent ontrat forms on supply hain oordination. This part of the literature is summarized in Cahon (2003) and Chen (2003). The ontribution of this paper is therefore to show that a supply ontrat, when properly designed and paired with an aution mehanism, an lead to an optimal prourement strategy (for one party of the supply hain). In other words, the paper highlights the role of a supply ontrat in a supply hain with supply side ompetition. Researh e orts to inorporate aution theory into the eld of operations management have been on the rise. Many papers have started to inlude an aution mehanism or some other prie-disovery mehanism in an operations ontext, see Mendelson and Tuna (2000), Gallien and Wein (2000), Lee and Whang (2002), et. Elmaghraby (2000) provides a survey of this area. Whereas our paper fouses on integrating a supply ontrat with an aution mehanism, others have studied the use of inentive ontrats in autions. See, e.g., La ont and Tirole (1987), MAfee and MMillan (1986, 87b), and Riordan and Sappington (1987). The setting typially inludes one prinipal and multiple agents. The prinipal has a projet, for whih the agents ompete. Eah agent possesses private information, and his ation is unobservable to the prinipal. The solution is to aution o an inentive ontrat among the agents. This part of the literature is also disussed in La ont and Tirole (1993). The rest of the paper is organized as follows. Setion 2 sets the stage by desribing optimal designs for single-unit prourement autions. Setion 3 takes up the optimal design issue for variable-quantity prourement autions. Setion 4 introdues a newsvendor model with supply-side ompetition, examines and ompares prourement strategies based on the autioning o of a supply ontrat or a xed quantity, and explores the linkages to the existing literature on supply ontrats and a prevalent pratie in the retail industry. Setion 5 onludes. 2 Single-Unit Prourement Autions Consider the following simple prourement problem. A rm is onsidering purhasing a single unit of a produt. The rm values the produt at p, whihmayrepresenttheprie that a ustomer is willing to pay for the produt, or if the produt is used as a omponent in the prodution of a nal produt, the value that is attributed to the omponent produt. 5
There are n, n 2, suppliers apable of produing the produt. The suppliers' osts of prodution are independent draws from a ommon probability distribution F (), 2 [; ], whih is di erentiable with F () = 0 and F () =1. Let i be supplier i's prodution ost, i =1; ;n. Eah supplier is privately informed of his own ost. We seek an optimal prourement strategy for the rm (also referred to as the buyer) that maximizes her expeted pro t, whih is the di erene between p and the prie paid for the produt in the event of a trade. If a trade does not our, her pro t is zero. The suppliers are risk neutral. The above prourement problem an be onverted to an aution design problem that is well understood in eonomis. 3 Imagine the following senario. The rm has already poketed p. The rm is the \seller" of the objet \the right and obligation to supply one unit of the produt to the rm," and the suppliers are the potential \buyers" of the objet. Let v i be supplier i's valuation of the objet, i =1; ;n. If supplier i gets the objet, he inurs a ost of i. Hene v i = i. The rm's valuation of the objet is v 0 = p, whihis how muh the objet is worth to the rm if it is not sold. This orresponds to the senario where no suppliers have been found to deliver the input and the rm has to \produe" the produt by itself at ost p, e etively returning the money it previously olleted. The rm's objetive is now to maximize its expeted revenue from selling the objet. This is an independent, private-value aution, with well-known optimal designs. The rm's maximum expeted pro t is p plus the maximum expeted revenue. Let F v ( ) be the umulative distribution funtion of v i, i =1; ;n.thus F v (x) =Pr(v i x) =Pr( i x) =1 F ( x); v x v where v = and v =. The aution is regular if J(x) def = x 1 F v(x) F 0 v (x) is inreasing in x (Myerson 1981). Note that J(x) = ~ H( x) where ~H(x) def = x + F (x) F 0 (x) : (1) Thus the above regularity ondition holds if and only if H( ) ~ is an inreasing funtion. This is learly true if F (x) F 0 (x) is inreasing in x or equivalently, F is logonave. Many probability 3 To the best of our knowledge, this type of onversion, although straightforward, has not been arefully doumented. As prourement problems attrat more and more researh attention espeially in operations management, it is useful to have the basi results in one plae for easy referene. 6
distributions are logonave, inluding the beta family, whih has the uniform distribution as a member, and the normal distribution trunated and saled to a nite interval. See Rosling (2002) for an extensive disussion on logonave probability distributions and further referenes on the topi. Throughout this paper, we assume H( ) ~ is inreasing. The optimal design for the above aution is well known. It alls for a reserve prie v, whih represents the minimum bid aeptable to the rm. If J(v) <v 0 <J(v), v satis es J(v )=v 0. Else, if v 0 J(v), v = v; andifv 0 J(v), v = v. The following ommon aution forms are all optimal: the English aution, the Duth aution, the rstprie, sealed-bid aution, and the Vikrey aution. For example, in the English aution, the objet will be sold to the highest bidder if the highest bid exeeds the reserve prie; no deal otherwise. In the Vikrey aution, the seller herself (i.e., the rm) submits a bid v. The highest bidder gets the objet and pays the seond highest bid. The rm's maximum expeted pro t is Z v p + v 0 Fv n (v )+n [xfv(x)+f 0 v (x) 1]Fv n 1 (x)dx (2) v where the rst term is the prie p the rm has already olleted, and the remaining terms represent the rm's expeted revenues in the aution (see equation (11) of Riley and Samuelson (1981)). Now return to the prourement setting. Let be the mirror image of v, i.e., = v. Reall that v 0 = p. Thus,if H() ~ <p< H(), ~ H( ~ )=p. Else, if p> H(), ~ = ; and if p< H(), ~ =. Still all the reserve prie. (The symbol will appear in many plaes throughout the paper. Its exat meaning should be apparent from the ontext.) The optimal aution design an be easily translated bak to the prourement setting. For example, in the English (prourement) aution, the suppliers openly bid down the prie they harge for delivering the produt, with the lowest bidder as the winner as long as the bid is not above. The rm's maximum expeted pro t, given by (2), an be written in a more intuitive form. Using v 0 = p, v =, v =, andf v (x) =1 F ( x), we an write the buyer's maximum expeted pro t as Z [p ~ H(x)]f (1) (x)dx (3) where f (1) (x) =nf 0 (x)(1 F (x)) n 1 def is the probability density funtion of C 1 =minf 1 ; 2 ; ; n g. ThereisatradeonlyifC 1 is below ; this explains the range of the integral. The rm pays the virtual ost, H(C1 ~ ), whih is the prodution ost of the winning supplier plus an 7
information rent. If F is uniform, then the virtual ost is 2C 1, with the information rent at C 1. As expeted, as the supply-side ompetition intensi es, the information rent dereases. In general, E[ H(C ~ 1 )] = E[C 2 ], where C 2 is the seond lowest prodution ost of the suppliers (see Appendix I for a proof.) That is, the expeted virtual ost when the rm does not post a reserve prie is exatly the expeted seond lowest ost. Of ourse, the reserve prie is a lever the rm an use to ontrol its expeted prourement expense, with the optimal reserve prie being a breakeven point, where the virtual ost equals the rm's valuation of the produt. 3 Variable-Quantity Prourement Autions We now generalize the single-unit prourement problem disussed in x2 toonewherethe rm (or the buyer) an purhase any number of units of the produt. Let Q be the purhase quantity. The rm an use these Q units of the produt to generate revenue R(Q). Assume R( ) is stritly onave and inreasing with R(0) = 0. As in the single-unit ase, there are n potential suppliers for the rm's input. But now, eah supplier has a onstant marginal prodution ost and an unlimited prodution apaity. Let i be the marginal ost for supplier i, i = 1; ;n. These osts are independent draws from a ommon probability distribution F over a nite interval [; ]. Eah supplier is privately informed of his own ost. We shall retain all the assumptions made earlier about F. In partiular, H(x), ~ as de ned in (1), is inreasing. We seek an optimal prourement strategy that maximizes the rm's expeted pro t. As before, the suppliers are risk neutral. 3.1 Quantity Autions Dasgupta and Spulber (1989/90), later as DS, provide an optimal strategy for the above variable-quantity prourement problem. It requires the rm to rst announe a ontrat P ( ), whereby the rm pays P (Q) forq units of input for any possible value of Q. Knowing P ( ), the suppliers eah name a quantity in a sealed bid. The supplier who bids the maximum quantity wins the ontrat, i.e., he is to produe and deliver his bid and, in return, reeives a payment from the rm aording to P ( ). (The other suppliers do not produe and do not reeive any payment from the rm.) DS show that with a properly hosen P ( ), the above aution implements an optimal diret mehanism that maximizes 8
the buyer's expeted pro t in a diret revelation game. The strategy is thus optimal, due to the revelation priniple. 4 We shall use quantity autions to refer to prourement strategies where the suppliers ompete by submitting quantity o ers. The DS design is therefore a quantity aution onduted in the sealed, high-bid format. We will also onsider quantity autions that use other formats. While DS fous on the diret-revelation-game formulation of the variable-quantity prourement problem, we will diretly address quantity autions. We believe a areful analysis of the sealed, high-bid ase is bene ial in helping us understand this prourement strategy. The derivation leading up to the optimal payment funtion helps us draw parallels with the single-unit problem and see how the onept of virtual osts omes about in the variable-quantity setting. Our analysis also unovers a tehnial issue overlooked by DS and provides an answer for it. Finally, by examining quantity autions under other aution formats suh as the English aution or the Vikrey aution, we gain deeper understanding of this type of strategy. Consider the sealed, high-bid, quantity aution. Given the payment funtion P ( ), the suppliers play a game of inomplete information (due to their private ost information), for whih the Bayesian-Nash equilibrium is an appropriate solution onept. Assume that there is a symmetri Bayesian-Nash equilibrium strategy; this is plausible sine the suppliers are ex ante symmetri (with iid osts). Denote this strategy by Q( ): a supplier with marginal ost bids Q(), 2 [; ]. Assume Q() is stritly dereasing in for 2 [; ], for some 2 [; ], and Q() =0for>. Note that there is a trade if and only if C 1,whereC 1 is the lowest marginal ost of the suppliers. Sine the lowest-ost supplier always wins the aution, we know R(Q(C 1 )) is the system's revenue and C 1 Q(C 1 ) the system's prodution ost. As a result, the expeted system-wide pro t is Z [R(Q(x)) xq(x)]f (1) (x)dx (4) where f (1) is the pdf of C 1. To derive the buyer's expeted pro t, it su±es to determine the suppliers' expeted pro ts. Take any i =1; ;n. Suppose all the suppliers but supplier i play the strategy Q( ). Consider the problem faing supplier i. InorderforQ( ) to be an equilibrium strategy, it 4 See, e.g., Kreps (1990). 9
mustbetheasethatsupplieri andonobetterthanbiddingq( i ), for all i 2 [; ]. In partiular, supplier i gains nothing by bidding Q(x), for some x 6= i,whihisthesameas playing the strategy Q( ) but pretending that his marginal ost is x. Take any 2 [; ], and suppose i =. Let ¼ i (x; ) be supplier i's expeted pro t if he bids Q(x) whilehis marginal ost is, given that all the other suppliers play Q( ). Note that ¼ i (x; ) =[P (Q(x)) Q(x)][1 F (x)] n 1 (5) where P (Q(x)) Q(x) is the supplier's pro t if he wins the aution (by bidding Q(x)) and [1 F (x)] n 1 is the probability of winning, whih ours if and only if every other supplier's marginal ost is greater than supplier i's \reported" marginal ost x. In order for ¼ i (x; ) to be maximized at x =, it is neessary that @¼ i(x;) @x j x= = 0, i.e., [P 0 (Q())Q 0 () Q 0 ()][1 F ()] n 1 [P (Q()) Q()](n 1)[1 F ()] n 2 F 0 () =0: Using this equation in the expression of ¼i 0(), where ¼ i() def = ¼ i (; ) forany 2 [; ], we have ¼i 0() = Q()[1 F ()]n 1.Setting¼ i ( )=¼ i ( ; )=0,wehave with ¼ i () = E[¼ i ()] = = = Z ¼ 0 (x)dx = Z ½Z Z Z ( Z x Z Q(x)(1 F (x)) n 1 dx (6) ¾ Q(x)(1 F (x)) n 1 dx F 0 ()d ) Q(x)(1 F (x)) n 1 F 0 ()d dx Q(x)(1 F (x)) n 1 F (x)dx: Beause the suppliers are symmetri, the sum of the expeted pro ts of the suppliers is simply n times the above expression. Subtrating ne[¼ i ()] from (4) gives the buyer's expeted pro t: Z [R(Q(x)) ~ H(x)Q(x)]f (1) (x)dx: (7) Note that H(C ~ 1 ) emerges as the virtual marginal ost of prourement for the buyer, just like in the single-unit ase. De ne Q (x) = argmax Q 0 [R(Q) ~ H(x)Q]; 8x 2 [; ]: (8) 10
Sine R( ) is onave and H( ) ~ is inreasing, Q (x) isdereasinginx. Set equal to the minimum x 2 [; ] withq (x) =0;ifnosuhx exists, set =. Itislearfrom(7)that the buyer's expeted pro t is maximized if Q ( ) arises as a Bayesian-Nash equilibrium in the game of inomplete information. Suppose Q ( ) is a Bayesian-Nash equilibrium. Take any 2 [; ]. From (6), ¼ i () = Z Sine ¼ i (; ) =¼ i (), we have from (5) Therefore, [P (Q ()) Q ()][1 F ()] n 1 = P (Q ()) = Q ()+ Q (z)(1 F (z)) n 1 dz: R Z Q (z)(1 F (z)) n 1 dz: Q (z)(1 F (z)) n 1 dz (1 F ()) n 1 : (9) Denote by P ( ) the payment funtion that satis es the above equation for all 2 [; ]. Notie that the above derivation for P is entirely based on neessary onditions in order for Q ( ) to be a Bayesian-Nash equilibrium. It remains to verify that under P ( ), Q ( ) indeed arises as suh. (DS did not address this issue.) To see this, suppose P ( ) isthe payment funtion, and assume all players but player i follow strategy Q ( ). Now in (5), replae P ( ) withp ( ) andq( ) withq ( ). We have ¼ i (x; ) =(x )Q (x)(1 F (x)) n 1 + Z x Q (z)(1 F (z)) n 1 dz; x; 2 [; ]: Note that @¼ i (x; ) =(x ) @ ³Q n 1 (x)(1 F (x)) : @x @x Sine both Q (x) and(1 F (x)) n 1 are dereasing in x, the partial derivative on the right side is negative. Therefore, @¼ i (x; )=@x > (resp., <) 0forx<(resp., >). Consequently, ¼ i (x; ) is maximized at x =. The following theorem, whih we attribute to DS, summarizes the above development. Theorem 1 (Dasgupta and Spulber) In the quantity aution de ned by the payment funtion P ( ) and onduted in the sealed, high-bid format, Q ( ) arises as a ommon Bayesian-Nash equilibrium strategy for the suppliers, and the buyer's expeted pro t is E[R(Q (C 1 )) H(C ~ 1 )Q (C 1 )]: This is also the highest expeted pro t the buyer an ahieve among all feasible prourement strategies. 11
Several observations are immediate. The amount of input the buyer atually purhases is Q (C 1 ),adereasingfuntionofc 1. Therefore, as ompetition intensi es, i.e., as n grows, C 1 beomes stohastially smaller, leading to a stohastially larger purhase quantity. Moreover, as expeted, the buyer's expeted pro t inreases with supply-side ompetition. To formally show this, de ne (x) =max Q 0 R(Q) H(x)Q, ~ a dereasing funtion of x beause H(x) ~ isinreasinginx. As n grows, C 1 beomes stohastially smaller, inreasing E[ (C 1 )], the buyer's expeted pro t. Finally, note that the e±ient trade between the lowest-ost supplier and the buyer, the one that maximizes their joint gains, is Q 0 (x) def =argmax Q 0 R(Q) xq. NotethatQ 0 (x) >Q (x) for all x. Hene asymmetri ost information auses supply hain ine±ienies by reduing the trade. This is reminisent of the well-known double-marginalization phenomenon: 5 : the marginal ost faing the buyer, i.e., the virtual ost H(C ~ 1 ), is higher than the system's marginal ost, whih is C 1. For independent, private-values autions, a well-known result is the revenue equivalene theorem, whih states that the autioneer is indi erent among many ommonly used aution formats suh as the English aution, the Duth aution, the rst-prie, sealed-bid aution, and the Vikrey aution. Does a similar result hold for quantity autions? In other words, if the buyer an freely modify the payment funtion to suit the aution format used, an she still ahieve the optimal expeted pro t by using an aution format other than sealed, high-bid? For example, the buyer an run a quantity aution in the Vikrey fashion: the suppliers submit quantity o ers in sealed bids, the winner is the highest bidder, but the quantity produed by the winning supplier and delivered to the buyer is equal to the seond highest bid. What is the buyer's maximum expeted pro t in this ase? The following theorem shows that the revenue equivalene theorem breaks down for quantity autions. The proof is in Chen (2003). Theorem 2 Quantity autions are sensitive to the aution format used: while the sealed, high-bid aution and the Duth aution are optimal (maximizing the buyer's expeted pro t), the English aution and the Vikrey aution are not. Moreover, the buyer prefers the English aution to the Vikrey aution. 5 See, e.g., Tirole (1988). 12
3.2 Entry-Fee Autions An important feature of quantity autions is that the buyer rst ommits to a payment funtion. This payment funtion represents a potential soure of revenue for eah supplier. This business opportunity (of trading with the buyer) is likely to be valued di erently by di erent suppliers, with the lowest-ost supplier ahieving the highest valuation. Therefore, an alternative way to selet a supplier is to ask them to bid in terms of an up-front, lumpsum fee. The winner is the supplier o ering to pay the highest fee. The winning supplier determines the prodution quantity (to maximize his pro t), delivers it to the buyer, and reeives a payment from the buyer aording to the payment funtion. We will use entry-fee autions to refer to prourement strategies where the suppliers ompete in terms of an entry fee. We shall show that entry-fee autions are also optimal for the buyer. Take any payment funtion P ( ) with P (0) = 0. Consider the problem of supplier seletion. De ne v() =maxp (Q) Q (10) Q 0 Let Q() = argmax Q 0 P (Q) Q. 6 Therefore, supplier i values the business opportunity at v( i ) def = v i, i =1; 2; ;n. Sine the suppliers' marginal osts are independent draws from a ommon distribution, the values fv i g n i=1 are independent and identially distributed random variables. Consequently, the problem of hoosing a supplier is like selling an objet (i.e., entry) to the highest bidder, where the bidders have independent, identially distributed valuations. From the revenue equivalene theorem, the buyer obtains the same expeted lump-sum fee (and selets the same supplier) if she uses the English aution, the Duth aution, the sealed, high-bid aution, or the Vikrey aution. Suppose the buyer uses the English aution for supplier seletion. That is, the suppliers openly bid on the fee they are willing to pay for the privilege to trade, and the supplier with the highest bid wins and pays his bid. Clearly, the supplier with the highest valuation (and the lowest marginal ost) wins the aution and pays (to the buyer) a lump-sum fee equal to the valuation of the seond-lowest-ost supplier. Let V k = v(c k ), k =1; ;n,wherec k is the kth lowest ost. Thus the lump-sum fee the buyer reeives is V 2. 7 6 This de nition is di erent from an earlier one, whih used Q( ) for a Bayesian-Nash equilibrium strategy in a quantity aution. 7 A reader versed in aution theory may think that the buyer an be made better o with a reserve prie. This is atually not the ase, as the subsequent development (summarized in Theorem 3) shows. The reason is that the aution is not regular in the sense of Myerson (1981). Chen (2003) illustrates this with an example. 13
We now proeed to determine the optimal payment funtion. Consider the buyer's ash ow. First, the buyer ollets a lump-sum fee, V 2, from the lowest-ost supplier. This supplier determines the trade quantity to maximize his pro t, i.e., maximizing P (Q) C 1 Q over Q. TheoptimalquantityisQ(C 1 ). The trade gives the buyer revenues in the amount of R(Q(C 1 )) but osts her P (Q(C 1 )). Consequently, the buyer's pro t is Sine V 1 = P (Q(C 1 )) C 1 Q(C 1 ), def = R(Q(C 1 )) P (Q(C 1 )) + V 2 : =R(Q(C 1 )) C 1 Q(C 1 ) (V 1 V 2 ): (11) We next obtain a onvenient expression for the expeted value of (V 1 V 2 ), whih is the winning supplier's pro t. Note from the optimization problem in (10) that Q() is dereasing in. (This is true for any P ( ).) Let 0 be the minimum with Q() =0. If Q() > 0forall 2 [; ], then set 0 =. Take any < 0. Thus Q() > 0. Writing v() =P (Q()) Q() and di erentiating, v 0 () =P 0 (Q())Q 0 () Q() Q 0 (): Sine P 0 (Q()) = (the rst-order ondition for the optimization problem in (10)), we have v 0 () = Q(). On the other hand, for any > 0, Q() =0andthusv() =0sine P (0) = 0. Consequently, for all > 0, v 0 () =0= Q(). Consequently, V 1 V 2 = v(c 1 ) v(c 2 )= Z C2 C 1 Q(x)dx: Using the onditional probability density funtion of C 2 given C 1 = (see Appendix I), we have " Z # C2 E Q(x)dx C 1 " Z # C2 = E C1 E Q(x)dxjC 1 = = = Z Z C 1 nf 0 ()(1 F ()) n 1 d Z y= Z µz y nf 0 ()d Q(x)dx y= µz y (n 1)F 0 (y)(1 F (y)) n 2 Q(x)dx (1 F ()) n 1 dy (n 1)F 0 (y)(1 F (y)) n 2 dy whih, after hanging the order of integration twie ( rst between x and y, and then between and x), beomes Z Q(x)nF (x)(1 F (x)) n 1 dx: (12) 14
Substituting (12) for the expeted value of (V 1 V 2 )in(11),wehave E[ ] = Z [R(Q(x)) ~ H(x)Q(x)]nF 0 (x)(1 F (x)) n 1 dx whih is exatly the same as (7), the buyer's expeted pro t in the high-bid quantity aution. (Note the di erent meanings of Q( ) in the two plaes.) Although the above expression is obtained under the assumption that the buyer uses the English aution to selet a supplier, we know, from the revenue equivalene theorem, that the same expression holds if she instead uses the Duth aution, the rst-prie, sealed-bid aution, or the Vikrey aution. Note that if Q(x) =Q (x),whihisde nedin(8),forallx 2 [; ], then E[ ] equals the buyer's maximum expeted pro t (see Theorem 1). This is indeed possible. The payment funtion that ahieves this, denoted by P ( ), is the solution to P 0 (Q ()) = (13) for all 2 [; ] withq () > 0. Note that this optimal payment funtion is inreasing in Q, onave beause Q () isdereasingin, and independent of the number of bidders. Also note that it is independent of the aution format used for supplier seletion. Finally, adding a onstant to the payment funtion does not hange its optimality. Theorem 3 The buyer maximizes her expeted pro t if she uses an entry-fee aution with P ( ) as the payment funtion. Moreover, this payment funtion is inreasing, onave, and independent of the number of potential suppliers. 3.3 Quantity vs. Entry-Fee Autions Although both quantity and entry-fee autions are optimal for the buyer, they are di erent along several dimensions. Under an entry-fee aution, the buyer has the exibility of using any of the ommon aution forms mentioned earlier. This exibility is lost for quantity autions (Theorem 2). Moreover, the optimal payment funtion for entry-fee autions is onave, inreasing, and independent of the number of bidders. But for quantity autions, it may atually derease (a rather unpleasant feature), and it generally depends on the number of bidders. In other words, the optimal payment funtion in entry-fee autions is more \intuitive" (a supplier is paid more if he delivers more) and more \detail-free" (the buyer does not have to know the exat number of potential suppliers). Finally, entry-fee autions t well with a ommon pratie in the retail industry, namely the use of slotting allowanes. More on this in the next setion. 15
4 A Newsvendor Model with Supply-Side Competition One of the most elebrated models in operations management is the newsvendor model. It aptures the essene of inventory deision making under demand unertainty. The newsvendor model assumes omplete ertainty on the supply side, where, typially, an unlimited quantity an be proured at an exogenously given per-unit wholesale prie. In reality, however, most industrial buyers fae multiple potential suppliers with private information about their prodution osts. As a result, the purhase prie needs to be disovered, whih of ourse in uenes the purhase quantity. Our objetive in this setion is to introdue a newsvendor model with supply-side ompetition and to haraterize optimal prourement strategies in this setting. A rm (newsvendor) must determine how muh inventory of a produt to stok in advane of a selling season. The total demand for the produt over the entire selling season, D, is a (nonnegative) random variable, with pdf g( ) anddfg( ). The selling prie is p per unit, whih is exogenously given. (Later, we will disuss the ase where the selling prie is a deision variable.) If the rm runs out of stok during the selling season, there are no replenishment opportunities and the exess demand will be lost. On the other hand, if there is exess inventory at the end of the season, it an be salvaged at v per unit. Let e be the proessing ost per unit of inventory, whih may inlude expenses suh as those inurred for spae and handling. On the supply side, there are n ( 2) potential suppliers for the produt. We shall retain all the assumptions made earlier about the suppliers in x3. That is, eah supplier is apable of produing an unlimited quantity of the produt at a onstant but supplier-spei marginal ost. Reall that i is supplier i's marginal ost, i =1; ;n,and these marginal osts are iid draws from a ommon probability distribution over the nite interval [; ], with pdf f( ) anddff ( ) satisfyingf () =0andF () =1. Eahsupplier is privately informed of his own marginal ost. Finally, the virtual ost funtion, H( ), ~ as de ned in (1), is stritly inreasing. The rm and the suppliers are all expeted-pro ts maximizers. We seek an optimal prourement strategy for the rm. Let Q be the level of inventory at the beginning of the selling season. The rm's pro t, exluding the osts inurred to purhase the inventory, an be expressed as p minfq; Dg + v(q D) + eq: Sine minfq; Dg = Q (Q D) +, the expeted pro t is R(Q) def =(p e)q (p v) 16 Z Q 0 G(y)dy:
For onveniene, we will refer to R( ) asthe rm'srevenue funtion. Clearly, this funtion is onave with R(0) = 0. For the rest of the setion, we will fous on the ase where the revenue funtion is stritly onave. Note that lim Q!+1 R 0 (Q) =v e, whihanbe positive or negative. Thus, unlike the revenue funtion used in x3, this one may eventually derease (in quantity). We assume p>v; p e> and e + >v (14) where the rst inequality is natural, the seond indiates that a pro table supply hain may exist, and the third eliminates any possibility for an arbitrage. 4.1 Supply-Contrat Autions It is easy to see that the problem faing the rm is idential to that faing the buyer in x3. 8 Therefore, the rm's optimal prourement strategy is to use either a quantity aution (the sealed, high-bid format) or an entry-fee aution (multiple formats). Reall that this requires the rm to rst o er a supply ontrat, i.e., a pay-for-delivery funtion, and then invite the suppliers to bid for the ontrat. Reall that Q ( ) is the solution to (8) (with the newsvendor revenue funtion). This is the rst step in determining the optimal supply ontrat. Note that the range of H( ) ~ is [; + 1 F 0 ( ], whereas R0 (0) = p e (assuming G(0) = 0). Sine p e>(see (14)), there are only two possibilities. If p e> H(x) ~ for all x 2 [; ], then Q (x) > 0forallx2 [; ]. By de nition, =. Otherwise, is the solution to p e = H( ~ ). Using the spei form of the newsvendor revenue funtion, we have Q (x) =G 1 ( (p e) H(x) ~ ); x 2 [; ] p v where G 1 is the inverse of the demand distribution funtion G. Clearly, Q (x) is dereasing in x. Let[Q; Q] denotetherangeofq ( ). Suppose the rm is onsidering using a quantity aution. The orresponding optimal supply ontrat is given in (9). Although there is in general not a losed-form solution 8 The only di erene is that the revenue funtion here may be unimodal. But this is learly inonsequential, as there is no loss of optimality if one restrits the prourement quantity to the range where the funtion is inreasing. 17
for P ( ), a simple iterative proedure an be used to ompute it by disretizing the interval [; ] and approximating the integral with a disrete sum. It is interesting to note that P (Q) is atually a quantity disount sheme beause P (Q)=Q, the average per-unit wholesale prie, is dereasing in Q. To see this, simply note from (9) that P (Q ()) Q () = + R Q (z)(1 F (z)) n 1 dz Q ()(1 F ()) n 1 and that the right side is inreasing in. (The proof, omitted here, does not require the spei form of the newsvendor revenue funtion.) Hene the optimal quantity aution amounts to autioning o a quantity disount sheme by inviting the suppliers to submit sealed bids in quantity spae. It is interesting that a quantity disount sheme is used even though the underlying prodution tehnologies are all linear. Bidding in quantity spae, however, is not the only way to implement the optimal quantity aution. It an also be done in the wholesale prie spae. De ne w(q) = P (Q) Q for all Q 2 (Q; Q). From the previous paragraph, w( ) is a dereasing funtion. Consider the following sealed-bid aution. Eah supplier submits a per-unit wholesale prie that they would harge. The supplier with the lowest wholesale prie wins the aution, and the amount the rm purhases from the winning supplier is the Q that makes w(q) equalto the lowest bid. It is straightforward to show that the Bayesian-Nash equilibrium in this aution, w () for 2 [; ], is suh that w(q ()) = w () for all, thus maximizing the rm's expeted pro t. Dasgupta and Spulber (1989/90) have made a similar point, albeit prediated on a di erent ondition. This type of aution, where the purhase quantity is a dereasing funtion of the prevailing wholesale prie, has also been studied by Hansen (1988), where the w(q) funtion is exogenously given. As we have shown in the previous setion, the rm's optimal prourement strategy an also take the form of an entry-fee aution. The orresponding optimal supply ontrat, P ( ), is given by (13). But here we have a losed-form solution. Take any Q 2 [Q; Q]. Thus there exists an x 2 [; ]withq = Q (x). Note that P 0 (Q) =x and R 0 (Q) = H(x). ~ Sine R 0 (Q) =(p e) (p v)g(q), we have P 0 (Q) = ~ H 1 ((p e) (p v)g(q)) wheretheinversefuntion ~ H 1 ( ) is well-de ned beause ~ H( ) is inreasing. Therefore P (Q) = Z Q Q ~H 1 ((p e) (p v)g(z))dz; Q 2 [Q; Q]: (15) 18
Of ourse, adding any onstant to the above still gives an optimal supply ontrat. Entry-fee autions t well with the use of slotting allowanes and vendor managed inventory (VMI), both of whih are prevalent in the retail industry. Slotting allowanes are lump-sum, up-front payments from a supplier to a retailer when the supplier introdues a new produt to the retailer's stores. Sometimes, suh payments are made in order to keep an existing produt on the retailer's shelves (these are also referred to as pay-to-stay fees). Therefore, a slotting allowane orresponds to the entry fee olleted by the newsvendor in our model. On the other hand, VMI is a pratie where the inventory replenishment deision is delegated to the supplier. In our model, this means that the supplier that has been seleted is given the deision right to determine the level of inventory to stok at the beginning of the selling season. Of ourse, the supplier should not be given omplete freedom in making the inventory deision. Instead, the deision must be guided by a ontrat, whih is the payment funtion in the entry-fee aution. It is interesting that in our framework, slotting allowanes and VMI arise as two important features of an optimal prourement strategy. The existing literature on the reasons for using slotting allowanes tends to fous on the signaling e et (when the supplier has more information about the produt's potential in the marketplae) or the risk-sharing e et (the retailer often inurs real, out-of-poket osts in new-produt introdutions and many new produts fail), see the FTC report (2001) and the many aademi papers on slotting allowanes suh as Sha er (1991), Chu (1992), Lariviere and Padmanabhan (1997), Sullivan (1997), Bloom et al. (2000), and Desai (2000). Our interpretation is simply that slotting allowanes are fees resulting from suppliers ompeting for sare shelf spae, and they an be part of an optimal prourement strategy for the retailer. This has the potential to add a new dimension to the debate on the purposes and onsequenes of slotting allowanes. Notie that in both the quantity and the entry-fee aution, the ontrat that the suppliers ompete for ontains no unertainty, i.e., the payment a supplier reeives depends only on the quantity it delivers to the rm. In pratie, there are numerous kinds of supply ontrats where the transfer payment also depends on the realized demand. 9 Let P (Q; D) be the transfer payment if Q units are delivered to the rm and the realized demand is D. When D is random, the transfer payment beomes unertain. Consider, e.g., the returns ontrat, whereby the rm pays a per-unit wholesale prie w for every unit of inventory stoked at the beginning of the selling season and if there is leftover inventory at the end of 9 See Cahon (2003) for a omprehensive disussion on various kinds of supply ontrats. 19
the selling season, the rm obtains a rebate of b per unit of exess inventory. 10 Under suh aontrat, P (Q; D) =wq b(q D) + : Another type of ontrat that has been observed in pratie is the revenue-sharing ontrat. Here the rm pays the supplier per-unit wholesale prie w for the initial inventory, and transfers 2 (0; 1) fration of the rm's revenues (regular sales and salvage sales) to the supplier. In this ase, P (Q; D) =wq + (p minfq; Dg + v(q D) + ): As the suppliers are risk neutral, their valuation of the supply ontrat is just E D P (Q; D) def = ~P (Q). If a supply ontrat an be identi ed so that P ~ ( ) =P ( ) (resp.,p ( )), then autioning it o among the suppliers in a quantity aution (resp., entry-fee aution) onstitutes an optimal prourement strategy. This is indeed possible, at least in some ases. Suppose F (x) =(x )=( ) for all x 2 [; ], i.e., the marginal prodution osts are independent and uniformly distribued. Hene H(x) ~ =2x, x 2 [; ]. From (15), P (Q) = Z Q 0 1 [(p e) (p v)g(z)+]dz: 2 (Herewehaveaddedaonstant, R Q 0, to the payment funtion. As noted in x3.2, the resulting ontrat remains optimal.) It an be easily veri ed that a returns ontrat with w = p e + 2 and b = p v 2 would have P ~ = P.Notethatbmay be higher than w, but this is ne beause the supplier hooses the quantity to deliver to the rm (in the entry-fee aution) and beause b + v<p (so the rm does not have any inentive to turn away demand). On the other hand, the above payment funtion an also be expressed as P (Q) = e 2 Q + p 2 E D minfq; Dg + v 2 E D(Q D) + : Therefore a revenue-sharing ontrat with w = e 2 and = 1 2 in onjuntion with an entry-fee aution, is optimal. Note that if w<0, then it represents a subsidy before revenue sharing. 10 Here the rm does not physially return the leftover inventory to the supplier, and it an still salvage the exess inventory for v per unit. 20
So far, it has been impliitly assumed that all the players (the rm and the suppliers) possess the same information about demand, i.e., they all know the distribution funtion G( ). It would be interesting to see the potential impliations of asymmetri information about demand. If the rm has better information about demand than the suppliers, then the rm should o er a ontrat that does not depend on the realized demand (i.e., P (Q)). This way, the suppliers do not fae any demand unertainty when evaluating the ontrat (and thus do not require distributional knowledge about demand). In ontrast, o ering a ontrat of the form P (Q; D) (e.g., a returns or revenue-sharing ontrat) would most likely lead to suboptimal results. On the other hand, if the suppliers possess better information about demand than the rm does (and the suppliers all agree on a ommon demand distribution, a strong assumption), then the \optimal" ontrat based on the rm's naive knowledge about demand is learly \seond-best." One way to get around this problem is to o er a ontrat of the form P (Q; D). And if the resulting P ~ = P or P regardless of the demand distribution, then the rm has ahieved the \ rst-best" outome, i.e., an outome that would result if the rm had aess to the suppliers' demand information. From the previous paragraph, we know that this ondition does hold in some ases. It is interesting that in suh ases, the rm an overome its informational disadvantage simply by o ering the right form of ontrat. We next onsider how the demand- and supply-side harateristis a et the rm's maximum expeted pro t (ahieved with an optimal prourement strategy). To this end, suppose D is normally distributed with mean ¹ andstandarddeviation¾. As this assumption implies the possibility of negative demand, 11 we modify the revenue funtion as follows: Z Q R(Q) =(p e)q (p v) G(y)dy: (16) 1 In this ase, we have Q (x) =¹ + ¾ 1 ( p e H(x) ~ ); x 2 [; ] (17) p v where ( ) is the df of the standard normal and 1 its inverse. Reall that is determined as follows. If p e> H(x) ~ for all x 2 [; ], then =. Otherwise, is the solution to p e = H( ~ ). Note from (17) that if <, thenasx ", Q (x)! 1. Thisisagain a onsequene of the normal demand assumption. (These unpleasant onsequenes of the normal assumption are routinely tolerated in the traditional newsvendor model.) 11 The likelihood of negative demand should be negligible in order for the normal assumption to be plausible. 21